You can do that. The combat phase is divided into several steps, and this is how the first step goes:
507.Beginning of Combat Step
507.1. First, if the game being played is a multiplayer game in which the active player's opponent's don't all automatically become
defending players, the active player chooses one of his or her
opponents. That player becomes the defending player. This turn-based
action doesn't use the stack. (See rule 506.2.)
507.2. Second, any abilities that trigger at the beginning of combat go on the stack. (See rule 603, "Handling Triggered Abilities.")
507.3. Third, the active player gets priority. Players may cast spells and activate abilities.
So, during the 3rd part listed there, you can cast Desecration Demon (if you have something that allows you to do so as though it had flash, such as Leyline of Anticipation). At this time, "At the beginning of each combat" have already been triggered, and it wasn't on the battlefield at that time, so it didn't trigger. By the time you cast it, it's too late for that trigger.
After that comes the declare attackers step, during which you can attack with your Demon if you have a card such as Fervor that will give him haste.